We propose a new numerical method for solving stochastic boundary-value problems. The method uses the deterministic method of lines to treat the time, space and randomness separately. The emphasis in the present study is given to stochastic partial differential equations with forced additive noise. The spatial discretization is carried out using a second-order finite volume method, while the associated stochastic differential system is numerically solved using a class of stochastic Runge-Kutta methods. The performance of the proposed methods is tested for a stochastic advection-diffusion problem and a stochastic Burgers equation driven with white noise. Numerical results are presented in both one and two space dimensions.